In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0,1)-matrix forbidding any (t+l)x(\lambda+2) submatrix (\lambda+2). (1_t 0_l) . Assume A is m-rowed and only columns of sum t+1,t+2,... ,m-l are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1,... ,t and m-l+1,m-l+2,..., m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t-design with block size t+1 and parameter \lambda and there are no other columns. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design based constructions. We extend in a few directions.

中文翻译：

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设计理论和一些禁止配置

在本文中，我们将 t-designs 与极值集理论中的禁止配置问题联系起来。让 1_t 0_l 表示在 l 0 的顶部的一列 t 1。我们假设 t>l。让 q。(1_t 0_l) 表示由 t 行 q 1 和 l 行 q 0 组成的 (t+l)xq 矩阵。我们考虑避免某些子矩阵的矩阵的极值问题。设 A 是一个 (0,1)-矩阵，禁止任何 (t+l)x(\lambda+2) 子矩阵 (\lambda+2)。(1_t 0_l) 。假设 A 是 m 行，并且只允许重复和 t+1,t+2,... ,ml 的列。假设 A 具有受给定限制的最大列数。假设 m 足够大。然后 A 的每列总和为 0,1,... ,t 和 m-l+1,m-l+2,..., m 恰好一次，并且给定适当的可分条件，总和 t+1 的列对应于块大小为 t+1 和参数 \lambda 的 t-design，并且没有其他列。该证明通过鸽巢论证推导出 A 的列数的基本上限，然后仔细论证，对于大的 m，将边界大量减少到基于设计的构造给出的值。我们向几个方向延伸。